Three-state intersection

For apparent reasons the most common type of surface intersections will comprise two states. Here we give a brief survey on their different topologies and symmetries. Three-state intersections (and higher ones) will be discussed below. 

B. The Study of a Real Three-State Molecular System Strongly Coupled (2,3) and (3,4) Conical Intersections... 

Appendix C On the Single/Multivaluedness of the Adiahatic-to-Diahatic Transformation Matrix Appendix D The Diabatic Representation Appendix E A Numerical Study of a Three-State Model Appendix F The Treatment of a Conical Intersection Removed from the Origin of Coordinates Acknowledgments References... 

It is important to emphasize that this analysis, although it is supposed to hold for a general three-state case, contradicts the analysis we perfonned of the three-state model in Section V.A.2. The reason is that the general (physieal) case applies to an (arbitrary) aggregation of conical intersections whereas the previous case applies to a special (probably unphysical) situation. The discussion on this subject is extended in Section X. In what follows, the cases for an aggregation of conical intersections will be tenned the breakable situations (the reason for choosing this name will be given later) in contrast to the type of models that were discussed in Sections V.A.2 and V.A.3 and that are termed as the unbreakable situation. 

We ended Section XV.A by claiming that the value a(r q = 0.4 A) is only 0.63ic instead of it (thus damaging the two-state quantization requirement) because, as additional studies revealed, of the close locations of two (3,4) conical intersections. In this section, we show that due to these two conical intersections our sub-space has to be extended so that it contains three states, namely, the second, the third, and the fourth states. Once this extension is done, the quantization requirement is restored but for the three states (and not for two states) as will be described next. 

In Section IV, we introduced the topological matrix D [see Eq. (38)] and showed that for a sub-Hilbert space this matrix is diagonal with (-1-1) and (—1) terms a feature that was defined as quantization of the non-adiabatic coupling matrix. If the present three-state system forms a sub-Hilbert space the resulting D matrix has to be a diagonal matrix as just mentioned. From Eq. (38) it is noticed that the D matrix is calculated along contours, F, that surround conical intersections. Our task in this section is to calculate the D matrix and we do this, again, for circular contours. 

In this exercise, we will examine a small part of this process. We will predici ihe relative energies of the three states at the ground state geometry, and we will locate the conical intersection. We ve provided you with an optimized ground state (cis) structure and a starting structure for the conical intersection in the files 9 06 gs.pclb and 9 06 ci.pdb, respectively. 

Although the importance of two-state conical intersections in nonadiabatic processes has been established [10, 26], the occurrence and relevance of accidental three-state... 

Figure 11-16. Diagram of the energy levels at the two- and three-state conical intersections in uracil and adenine calculated at the MRCI level, ciIJK represents conical intersection between states Sj, S j, S -. (From Ref. [210])...

Han S, Yarkony DR (2003) Conical intersections of three states. Energies, derivative couplings, and the geometric phase effect in the neighborhood of degeneracy subspaces. Application to the allyl radical. J Chem Phys 119 11562... 

Coe JD, Martinez TJ (2005) Competitive decay at two- and three-state conical intersections in excited-state intramolecular proton transfer. J Am Chem Soc 127 4560... 

Coe JD, Martinez TJ (2006) Ab initio molecular dynamics of excited-state intramolecular proton transfer around a three-state conical intersection in malonaldehyde. J Phys Chem A 110 618-630... 

Matsika S, Yarkony DR (2002) Accidental conical intersections of three states of the same symmetry. I. Location and relevance. J Chem Phys 117 6907... 

Matsika S, Yarkony DR (2003) Beyond two-state conical intersections. Three-state conical intersections in low symmetry molecules The allyl radical. J Chem Soc 125 10672... 

Matsika S (2005) Three-state conical intersections in nucleic acid bases. J Phys Chem A 109 7538... 

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